COMBUSTION AND FLAME 26, 151-162 (1976) 151

The Quenching of Deflagration Waves

J. BUCKMASTER Mathematics Department and Department of Theoretical and Applied Mechanics,

University of lllinois, Urbana, Illinois 61801

A simple model of a low Mach number deflagration wave is studied, with emphasis on the effects of a phenomenological heat loss term. It is shown by explicit construction that two quite different solutions, a fast wave and a slow wave, are possible in the adiabatic limit. In both cases the chemical reaction goes to completion and all the fuel is consumed, but for the fast wave the temperature increases from the ambient value to the usual adiabatic flame temperature, whereas the slow wave is isothermal. An analysis for finite heat loss is carried out in the realistic limit of infinite activation energy. In general this leads to a much simplified numerical problem, but for values of the flame speed of greatest physical interest an analytical description is possible. This predicts quenching when the heat loss is too great or the reaction rate too small, and yields simple explicit quenching criteria.

1. Introduction It is well known that a laminar diffusion flame can not propagate in a mixture confined within too narrow tubes, a result generally attributed to heat losses through the walls. The problem has been widely discussed starting with the pioneering work of Spalding [7], who describes an approximate analysis. Numerical computa- tions by Adler and Spalding [1] and others (e.g., Gerstein and Stine [3]) confirm the qual- itative predictions of the earlier study, the most important of which are that if the heat loss is too great the flame is quenched and there is no steady-state solution; and for heat losses less than the quenching limit there are two so- lutions, a fast wave and a slow wave. Thus the response diagram sketched in Fig. 2 is charac- teristic of the problem. The fast wave is appar- ently stable and is the one usually observed. Spalding and Yumlu [8] reported observations of the slow wave in a specially constructed stabilizing apparatus, and although questions were raised about the accuracy of their heat loss measurements, Yumlu [9] sustained the view that the slow wave is observable. Never- theless, the slow flame is not of great physical interest.

In view of the long and illuminating history of this problem, the reader may wonder why yet another analysis is required, particularly since as we shall see, few new qualitative insights are forthcoming. What distinguishes the analysis of the present paper from earlier analytical work is that after a clearly defined set of model equations is written down, their solution is dis- cussed in a rational way without recourse to ad hoc approximations. One of the tiroblems with a complicated subject such as combustion is that even after simplified model equations have been postulated, analytical results are usually only possible after a great many, often ques- tionable, additional approximations have been adopted. One then has the challenge of decid- ing whether the results are creatures of the model equations or of the analysis. Small won- der, then, that importance is attached to num- erical confirmation of any predictions. The present analysis, being rational, is an accurate substitute for the numerical work and indeed we essentially duplicate most of the results of Gerstein and Stine [3]. Of course our results have all the familiar advantages that a simple formula has over numerical output. Moreover, the present work lays the founda.tion for

Copyright ~1976 by The Combustion Institute Published by American Elsevier Publishing Company, Inc.

152

analyses of model equations more general than those adopted here. Suppose for example, one wanted to simulate the effect of wall capture of hot radicals (Simon, Belles, and Spakowski [6]). This could be done by adding an appro- priate sink term to the species equation and then carrying out an analysis similar to the present one.

The key to the present analysis is the adop- tion of Arrhenius kinetics characterized by an activation temperature that is much larger than the adiabatic flame temperature. An asympto- tic analysis is then possible. Activation energy asymptotics has only recently been exploited to any significant extent, and has proven to be an extremely powerful tool in a variety of prob- lems. For example, Bush and Fendell (1970) solved the problem of adiabatic deflagration waves in this way, thereby rendering obsolete a large amount of earlier work in the subject. We perform the same service for the non-adiabatic case.

The problem is discussed in several stages. After the formulation in §2 the fast-and slow- flame adiabatic limits are described in §3. The fast-flame limit can only be analyzed when the activation energy is large, and this is Bush and Fendell' s problem, but the slow-flame limit can be discussed without any such restriction. This analysis brings out the striking physical differ- ences between slow flames and fast flames.

Section 4 is devoted to activation energy asymptotics for the special case of a linear de- pendence of the heat loss on temperature. Only a brief outline of the analysis is given since it is only needed if a complete description of the slow-flame branch is required, and further- more, analytical results are not possible. The analysis of the fast-flame branch is carried out in §5 when the heat loss is an arbitrary function of temperature. This section contains the major results of the paper, including explicit quench- ing criteria.

2. Formulation Williams [10] formulates a mathematical prob- lem for adiabatic deflagration waves and we shall follow the formulation closely with some generalizations, the most important of which is the addition of a heat loss term to the energy

J. BUCKMASTER

equation. A one-step reaction mixture --> product is assumed so that the combustion field is characterized by two variables, T the tem- perature and Y, the product mass-fraction. The characteristic Mach number of the flame is as- sumed sufficiently small for the pressure to be sensibly constant. A constant specific heat Cp is assumed, but the heat conduction coefficient k is allowed to vary with temperature. This makes it expedient, following Williams, to de- fine an independent space variable by

X

~ = m c p f dx X '

where m, the mass flux through the flame as observed in a frame in which the flame is sta- tionary, is a constant that must be determined. Then

d Y _ Le [Y- e], (2.1) d~

where Le is the Lewis number, assumed con- stant, and e is the product flux fraction as de- fined by Williams. Changes in E are governed by

de _ 602

dx m

where to2 is the mass of product generated per unit volume per unit time, so that we shall write

de _ B(7)_ (1 - Y)exp - , (2.2) d~ rn 2 \ R O T

Where T" is the adiabatic flame temperature defined by

T'--T + ( h m ° - h P ° ) c

In this expression T= is the ambient tempera- ture, hm ° is the heat of formation of the mix- ture and hp ° is the heat of formation of the

THE QUENCHING OF DEFLAGRATION WAVES

product. Defining a new temperature variable by

h o o T=T~+( - m - h p ) 7",

c.

Eq. (2.2) may be written in the form

de = O=gZ×(r) (I - 1I) e x p [ ~ l , (2.3) d~ a'" +r

where 0 is a non-dimensional activation energy defined by

E1 0 -

go~"

and

(hal ° - he ° ) Or=

f~ is a constant that is a measure of the reaction rate divided bym z, and in recognition of this we define to' by

t

m 2.

is an arbitrary function that accounts for the preexponential temperature dependence of the reaction rate and also includes a contribution from h(T).

The energy equation may be written in the form

d 2 T - dr de + K¢(T) , (2.4) d~ 2 d~ d~

where

X~oO-) = x o

m2 G (hal° - hP ° )

and Q is the heat loss per unit volume per unit time. ~o(r) is an arbitrary function (except for

153

the restriction ~o(0) = (0) and thus permits an arbitrary dependence of Q on z. Note that the constant K is a measure of the heat loss divided by m * and in recognition of this we define Q' by

K=Q' m 2

Eliminating ~ leads to the equations that we shall solve in this paper, namely

Le d 2 Y _ d Y _ 0 2 ~ x ( r ) (1 - Y) e x p [ ~ ] , d~ 2 d~ cC'+r

(2.5a)

and

d2T _ dr d ~ 2 d~

+ K~(T)-0 2 ~2X(r) (1 - Y)exp[ 0 r--(LzD ] 0~'1 + T

(2.5b)

Boundary conditions appropriate for this sys- tem are

~_oo r ~ 0 , Y ~ 0 , (2.6a)

-++oo (i) K 4 : 0 : r ~ 0 , d~Y~0, d~

(2.6b)

(ii) K = 0 : r ~ 1, Y-+1.(2.6c)

It is apparent that to resolve the cold bound- ary difficulty it is necessary to choose either a 1 --- 0 or ×(0) = 0. Then the reaction rate

vanishes as I ~ [ -+ ~ when K 4= 0, since r then vanishes far from the flame. One consequence of this is that when there are heat losses we cannot infer that Y ~ 1 as ~ ~ + oo. Indeed, as the subsequent analysis will reveal, there is in general a residue ofunburnt mixture left behind by the flame.

This conclusion is different from that of Gerstein and Stine [3] simply because they as- sume a reaction rate proportional to

E l (1 - Y) exp (-- ),

ROT

154

and T :/: 0 in their calculations so that the only way this expression can vanish as ~ -~ + 0- is if Y ~ 1. The present formulation, which per-

mits the indefinite existence of a cold unburnt mixture, is clearly more realistic.

It is well known (Johnson [4]) that mere reso- lution of the cold boundary difficulty does not guarantee a unique flame speed for the adiaba- tic problem; in general there is a continuum of possible flame speeds. However, by introduc- ing a cut-off temperature, slightly larger than the cold-boundary temperature, and below which the reaction rate is zero, uniqueness is assured [4 ]. Similar difficulties might occur for the non-adiabatic problem but can presumably be handled by a suitable choice of the behavior of ×(r) near r = 0 (cut-off corresponds to X =0 when r < r * , w h e r e 0 < r * < l ) .

Explicit choices do not have to be made when the activation energy is large, since the solution is sensitive only to the behavior of the reaction rate near the temperature maximum.

We shall analyze the system (2.5)-(2.6) by supposing that K, o:, 0 and Le are specified constants, ¢ and X are prescribed functions, and II is a constant that must be determined as part of the solution. Thus ll is the basic eigen- value of the problem.

3. Adiabatic Limits Adiabatic deflagration corresponds to Q ' O , i.e.,

Km2 = 0. (3.1)

The classical adiabatic limit is attained when K = O and this is the problem analyzed by Bush

and Fendell [2] in the limit 0 ~ *o.Their analysis requires consideration of the combustion field in three distinct regions. In~ < 0 the chemistry is frozen; in //> 0 chemical equilibrium pre- vails' so that Y = 1 correct to all algebraic order in 0; between these t w o regions there is a fame-sheet which smooths out the gradients and in which appropr ia te var iables aren = O$,y(rl) = 0 (Y-l), t(r/) = 0(z-1).Solutions in these three regions are easily constructed and matched with each other and in this way the following expression for the flame-sheet is deduced,

m = [26o' X(1) Le (1 +a")21 '/2. (3.2)

J. BUCKMASTER

The limit corresponds to m ~ constant, Q' --* 0 when to' is fixed, or m ~ 0% co' ~ oo when Q' is fixed. The temperature profile is sketched in Fig. 1.

An alternative adiabatic limit comes from the choice K --, oo for this corresponds to m ~ 0 in such a way that, in the limit, Eq. (3.1) is satis- fied. Crude ad hoc analyses (e.g., vonKarman, see Williams [10]) have indicated something of the nature of the solution in this limit, but a rational asymptotic description is possible for large K. We shall sketch the details for the special case

~ = r , x = r 3', a =°°. (3.3)

It is appropriate to write 3+_~

0 2exp(0)[2=K2 2 ~ e x p ( O ) , (3.4) T ,

where z,(K) is the maximum value of z, as- sumed located at the origin, and ~ is an O(1) quantity.

On a scale~ = O(l)the reduced equations, correct to all algebraic orders in K, are

r = 0 (3.5a)

1 d 2 Y _ d Y Le d~ 2 d~

(3.5b)

and a solution for Y is sought in the form

where the superscript (+) refers to ~ > 0, (-) to ~ < 0. To avoid subsequent contradic-

tions in the analysis we are compellec] to

make the choice

+ Yo =1.

+ Moreover, Y1 = B, a cons tan t .

On a scale ~ = O(~K-K ) , appropriate vari- ables are

1 I.e., if fuel were present it would be instantly con- sumed.

THE QUENCHING OF DEFLAGRATION WAVES 155

I I

I l ~ Flome-sheet of

thickness 0(I/8)

Fig. 1. Tempera ture profile for an adiabatic flame (K = 0).

I"

E I I I

n = v ~ , t(n)=x/-g~, t ,=V-g-~, ,

(3.6)

y(n) = x/K ( r - 1),

and the equations, correct to all alebraic orders in K when ~ = O(1) are

1 d2y _ 1 dy Le &72 ~ dT?

d 2 t _ t + 1 dt drl 2 ~ drl

Solutions are sought in the forms

y+ ++-+ 1 + 1 ~Yo - ' ~ Y7 + 0 ( )

+ e-k-

and amongst other results,

+ Yo =B.

Finally, an inner region (a flame-sheet) is described on a scale ~j---O(--L-1 )in terms of the variables K

~" = x,/K~, s(~') = V ~ (t - t,), z(~') = x/-K-(.v-B), (3.7)

and solutions are sought in the form

s~s o + ~ _ ~ s ,+O( ),

1 z I + O ( ~ - - ) Z~Zo + vI- ~

--~ +--_L t , + O ( 1 ) . t , t , o V ~

Construction of the solution in each region and matching is essentially straightforward and yields the following results:

(. ~ 1 + 1 1 Le t , 2 ~ 4 0 ~ +O( ,(3.8)

s - te (3.9) 20 '

3+_7_ ~2 ''.22+3' K 2 2 exp[20Vr~-+Le_ 1].

Le (3.1o)

156

Clearly the limit corresponds to m ~ 0, Q' ~ 0 if ~ ' is fixed, or m ~ 0, co' -~ oo if Q' is fixed.

Note that B is a measure of the amount of un- burnt mixture left behind by the flame. This

J. BUCKMASTER

res idue is O~/~/ ' -~] and van i shes in the limit.

The results of this section are consistent with the familiar response diagram, as sketched in

m

Adiabatic Limit

Adiabatic Limit

Fig. 2. Flame speed as a function of heat loss, ~ fixed.

~ E

m

~FFas~-flame J Adiabatic Limit

C J Q u e n c h i n g

~ B

Slow -flame Adiabatic Limit >to

Fig. 3. Flame speed as a function of reaction rate, E fixed.

THE QUENCHING OF DEFLAGRATION WAVES

Figs. 2 and 3. The points A correspond to the result (3.2) and the points B to (3.10).

Figure 4 is a sketch of the flame structure in the slow-flame adiabatic limit. The reaction goes to completion in the limit and the profile for Y is indistinguishable, on the ¢ scale, from that in the fast-flame limit outside of the flame sheet. However in striking contrast to the fast- flame limit, the structure is isothermal. Of course on a physical scale the slow flame is much thicker than the fast flame.

Y,-r 0 ( I A / -K )

i o( /K) I I -..~ ~ [ Y= I

, i i ' ]

! L F l o m e - sheet

l y T h e r m o l D i f f us ion Zone

i

,

Fig. 4. Slow-flame structure in the adiabatic limit.

4. Activation Energy Asymptoties, K = O(1) In order to bridge the gap between the limits of small K and large K, an analysis can be carried out in the asymptotic limit 0 ~ . In this section we outline such an analysis when K = O(1).

On of the keys to a large 0 analysis is the ability to explicitly solve the frozen equations (i.e., when 1) = 0). WhenK = O(1),the function ~(r) generates difficulties in this respect, but these difficulties are eliminated if ~0 is a linear function. Thus, in this section we choose

and moreover (although this is not necessary) we adopt the other choices defined by (3.3).

It is apparent that r assumes a maximum at some point, and with no loss of generality this may be located at ~ = 0. Write

A / t ~ \

~2 0 2 e x p ( 0 ) : L~-exp[ u--I, (4.1)

157

where r* is the m a x i m u m tempera tu re and

7", ~ r , o +6"r, 1 +O(6Z).

= r , ° 2/0 is the small parameter of the problem. The governing equations then have the form

1 d 2 y _ d Y A 7") , ( l_Y)exp[ 0 ( r _ r , ) ] , Le d~2 d~ 62 7"7-,

d 2r _ dr +Kr_~6 A- rT(l-Y) exp[ 0_(r_r,)] ~ d~ 7"7", '

and except in the neighborhood of the origin the combustion is frozen in the limit 0 --9 ~. In the frozen regions a solution is sought in the form

There is a flame-sheet of thickness 0(8) cen- tered at ~ = o and the appropriate expansions within this sheet are

~ = 6r/,

Y ~ 1 + 6y , ( r t )+ 0 ( 6 2 ) ,

r ~ r , + 8r~ 01) + 0(6 z),

A ~ A o + 6 A 1 +0(62) .

Matching with the solution in the frozen re- gions leads to the following problem for the flame-sheet structure:

d2 tl + 7 0 - ~2o Le 7",04 exp(0 - - - - - + T , 1

dr/2 r , 0

158 J. BUCKMASTER

+ t l ) X [tl - (1 -x / l+4-K- ) r / - W] (4.2a) 2x/1+4K

t~ (0) = t' 1 (0) = 0, (4.2b)

t ' 1 (oo) = 1 - N / ~ ' - ~ 2 lx/q-+-4K =- u+' (4.2c)

, 1 + Vr]-+4K t l (_oo)= 2 l + N ~ .~//- , (4.2d)

t ~ / a -+~+p++O(1) as r~-~_+oo, (4.2e)

where

r , 0 = (1 + 4K) -½ (4.20

and

7",, (1 + 4 K ) = - Le W + p- -

+ [_1 - p + ~ ~ - Le]. (4.2g) 2 2

Since four boundary conditions are given for the second order equation its solution implies the s i m u l t a n e o u s d e t e r m i n a t i o n o f W(K), p+-(K) and

Le r ,o 4 + 7 ~2o exp(- 0 T~, +r*l)-~f(K)" (4.3) 0

is then known from (4.2g) as a function of K and the Lewis number, and fi- na l ly (4.3) de t e rmines £2 o as a func t ion of K, 0, Le, and 3'. In principle then, the prob- lem is c o m p l e t e l y so lved once the four • + functmns W, p- a n d f have been determined.

It is important to note that

e -0 (Ao + r A I +.. .) exp[ 0 ]

7", 0 7",0 + 67", 1 + . . . .

(thusS2o =-A°4 exp[- 0 - 0 - r , 1 ] ) , T~O 7"~0

and the variation of 1-/ with K is essentially determined by the factor exp (O/Z,o). It would be inconsistent in computing the derivative of lq to include the lower order terms that arise by

A0 differentiat ing z , 1 or - - unless the term

7",04

proportional to is also retained. Now

dn~ "" I 20 -in f2 dKn L(I + 4K)~J and since fl is positive it follows the ~ and all its der iva t ives are m o n o t o n e , decreas ing functions of K. Furthermore, as K ---> O we have from (3.2)

f2 ° ~ 1 , 2Le

whereas (3.10) shows that 3 + Y _

~2o o~K 2 2 exp(20x/~-) as K - oo

Thus i f Q ' , 0 , L e and Tare f ixed the graph ofI~0vs m has the form shown in Fig. 5. Where it intersects the graph ofco'/m2(if it does) defines two possible flame speeds. Clearly if Q' is too large or to' is too small the curves do not inter- sect. Thus, provided the problem (4.2) has a solution it is apparent that it will generate re- sponse diagrams of the form sketched in Figs. 2 and 3.

Explicit construction of the response re- quires numerical computations which we have not performed since O(1) values of K corre- spond to the slow-flame branch. 1 The fast- flame branch corresponds to O(1/0) values of K and then the problem can be solved without recourse to the computer. This analysis could continue from (4.2) but the results would then only be valid for the special choices (3.3). To solve the general problem it is necessary to start with the choice K = O(1/0) a priori, and this the point of departure of § 5.

In § 3 it was found that the large K there is a residue of unburnt mixture behind the flame• The present analysis for finite K reveals that behind the flame (~ > 0 )

1More precisely, to that part of the response for which the flame speed is exponentially smaller than (3.2).

THE QUENCHING OF DEFLAGRATION WAVES 159

z~m Jb')

i

l ~ ~ ~ E c o n s t a n t

Slow- -- * Fo st - flame flame

~ m

The boundary condition r(o~) = 0 arises sol- ely because of heat losses, since in the adiaba- tic case the temperature does not decrease be- hind the flame front and the downstream boun- dary condition is then r(~) = 1. Small heat losses cause only a weak decay in temperature behind the wave and significant deviation from r = I occur only on the large scale ~ = O(0), ~> O. We shall not consider the structure on this scale so that the solution that we shall describe in ~ > 0 is a small perturbation about z = 1.

Limiting equations valid when 0 ~ o~ maynow be written down. When ~ is negative and O(1), (r - 1) is a negative O(1) quantity so that the reaction term is transcendentally small in the limit. Then to all algebraic orders the equations

Fig. 5. Demonstration that there are either two solutions or none in the limit 0 ~ co

Yo ~ 1 - 1 0 L e ( W - p + ) + O ( ~ 2 ) (4.4)

~<0 1 d 2 Y _ d Y

Le d~ ~ d~

and (5. ! )

so that there is O(1/0) residue. Both Wand p+ Vanish as K ~ 0.

5. Activation Energy Asymptotics, K = O( 1 )

In this section the problem posed by (2.5), (2.6) is analysed in the limit 0 ~ oowhen K = O(1/0). It is helpful to write

K =__~_K , 0

dRr _ dr + K (p(r), d~ 2 d~ 0

describe this frozen region. When ~: is positive and O(1), (r- 1) is an O(1-)

i

quantity so that the factor 02determines the nature of the limiting equations. In the limit the region is an equilibrium one, and

Where/~ is O(1), and then the analysis is rela- tively straightforward once certain key facts have been established.

It is clear from the boundary conditions that r assumes a maximum, and with no loss of gen- erality this maximum may be located at ~ = 0. In the absence of heat losses (K = O) the maximum temperature is the adiabatic flame temperature (r = 1) so that it is not surprising that in the general case for which the heat los- ses are O(1/0),the maximum value of r differs from 1 by O (1/0). This is the motivation for writ- ing the chemical production term in the in the form (2.2).

~>o Y = 1 + (transc.)

d ~ = d~ + ~ ~(~) d~ 2 d~ 0

(5.2)

Between these two regions there is a thin flame-sheet located at ~ = 0 in which there is essentially a balance between the diffusion and reaction terms. The primary mathematical role of this layer is to smooth out the first deriva- tives which on a scale~ -- O(1) are discontinuous at~ = O. The solution in the sheet has the form

160 J. BUCKMASTER

1 0+ 7- - 7-, + ~- t, (~) + t= (~) + . . .

where r/= 0 ~, and r , is the maximum value of r, (achieved as 17 -+ o% as is clear from the equation to t~, be- low) and

7- ,~1 + ~ - r , + 0 ( ).

The variables ta, and y~ satisfy the equations

1 d 2 y l _ dZtl

Le d@ dp 2

~ t l +7-,~ - ~ o X ( 1 ) y I e x P l - - I

L l+a"j (5.3)

where f~oiS the first term in the asymptotic expansion of f~. We make the choice

Yl = Le tl (5.4)

so that

dr1 2 t~ r t , l + t l l (-d~-) = 2aoX(1) E e l dt , ( t , ) exp

(5.5)

This last expression is a first integral of (5.3), (5.4) subject to the condition

dtx tl - + 0 , - - ~ 0 as ~ o o . (5.6)

dr/

Of particular importance of its value as "0 '~ -~ (tl --+ -~)

dtl (-~)1 2 = 2 g 2 o X ( 1 ) L e e x p F r * l - ] ( l + a " ) 2. 1

(5.7)

Solutions in the outer regions, that is the regions governed by (5.1) and (5.2), have the form

1 y~+ --ra + - g . . .

1 rT+ 7---7-0+ 7 . . .

+ 1 r + + 7 ~ 1 + - - . . . 0

Where the superscript (-) refers to the reg- ion~ < 0, (+) to the region~ > 0. General solu- tions are easily constructed for arbitrary func- tions 0(r) and these must match with the flame sheet solution and satisfy the boundary cond i t i ons (2.6a) as~ -~ _oo. The con- dition r(~) = 0 can not be invoked because of the non-uniformity when ~ is O( 0 ) mentioned in the introductory remarks to this section. It is replaced by the requirement that r diverges at most algebraically as ~ -+ + ~.

In this way we find

Yo = e~ Le, ro = e ~. (5.8)

Matching of gradients with the flame-sheet so- lutions then implies, because of (5.7),

(1 +a-1)-2 -7-* 1

~ 2 0 - e x p [ 1-~-~a.1 ] (5.9) 2X(I) Le

so that tl is uniquely defined and has the asymptotic behavior

as r/-~_o~ tl ~ r7 + #7 + (transc.), (5.10)

where**7 is a known constant. Continuing,

7--, =~e~ f dge- 7 d~"c~(e ~ )+(7-, +#~)e ~,

(5.11)

Yt = Le U/" e~Le, (5.12)

and r + = - / ~ (1)~ + r , t . (5.13)

Further calculation, in particular the deter- mination of r . ~ to complete the definition of ~o, requires an examinat ion of the third-order flame-sheet structure (i.e., y2, tz). y2 and t2 are related through the equation

THE QUENCHING OF DEFLAGRATION WAVES 161

1 __d2y2- ~=dyl d=t2_. --,dtl (5.14)

L--~ drl = drl drl 2 drl

and furthermore they must have the following asymptotic behavior:

.q ~ + oo Y2 ~ (transc.)

+ + t2 ~ P2 r~ + ~z2 + (transc.),

77 -+ _~o Y2 ~ -1- Le2 r?2 + a 2 r/+ 3~+(transc.) 2

1 ~/2 + p~ ~ + / ~ + (transc.). t~ ~

+ + p~-,/~-, c~ and 3] are unknown constants at this point, but because of (5.14) they are related:

+ Pa = - - - + Le ~t~ + P~ -/l~. (5.1 5)

Le

Moreover , higher-order matching with the outer solutions leads to the results

p+=-EeO),

0 +/11+Kf o(e

and

a~ = Le 2 /J~.

Substituting these into (5.15) determines r , ,

I J 1 r , =-K" ~o(1) + ~o(e~)d~ , (5.16)

and then (5.9) may be written in the form

0

Ol,o(1)+ f I c o ' _ ( l + o i ' ) "2 exp(Q_~ -= ~. m 2 2X(1)Le m 2 (1 +a - l )

(5.17)

Equation (5.17) is the central result of the present paper. It reduces to the fast-flame adiabatic limit (3.2) when Q' vanishes.

When o~'is fixed, m is a multiple-valued func- tion of Q ' provided Q ' is not too large, and when Q' is fixed m is a multiple-valued function of J p r o v i d e d J i s not too small. Quenching conditions are easily deduced. When eo]s fixed

am_ [ z Q ' _ I I _ z dQ' m 2m '

where ~ -=- 0

0

[~o(1) + f ~p(e~)d~J

(1 + ~'1 )

so that the quenching poing (c in Fig. 1) has coordinates

m : [ 2 w ' X ( 1 ) L e ( l + c ~ ' l ) 2 e ' l ] ½, (5.18a)

Q ' = 2 w X ( 1 ) L e ( l + a q ) Z e "~y~'t. (5.18b)

Similarly, when Q is fixed the quenching point (C in Fig. 2) is

m = x / ~ 7 , co, = e~Q' (5.1.9) 2X(1) te (1 +a-~) 2"

I f

Q'>2~'X( l )Le( l+o~ ' l )2eqZ -~ (5.20)

there is no solution to the problem. It is noteworthy that the flame speed at ex-

tinction is e 1/2 (----0.61) times the adiabatic fast-flame speed, a result which is independent of all physical parameters. Gerstein and Stine [3] suggested such a relationship on the basis of their numerical computa t ions , assigning a value 0.54 to the ratio. Their results are for an activation energy of 40,000 cal/mole and an adiabatic flame temperature of 1545 ~-K, data consistent with large activation energy asymp- totics(0 ~ 13),so that the agreement is not for- tuitous.

In addi t ion it should be noted that the maximum temperature at extinction is only

162 J. B U C K M A S T E R

slightly smaller than the adabatic flame tem- perature. Indeed, along the whole of the fast- flame branch the difference is O(1/0).

There has been some debate on the question of the effects of heat losses upstream of the flame front (~ <O)relative to losses downstream ([ > O) (e.g. Williams [10,p. 191]). This was

perhaps prompted by the analysis of Spalding [7] who neglected upstream losses. The present results distinguish be tween the two con- tributions. Upstream losses make a contribu- tion represented

0

by the factor f ~o(e~)d ~, whereas downstream ° ~

losses are represented by ~o (1) (Eqs. (5.16), (5.17)). These contributions are equal if ¢ is proportional to z (which is appropriate when the losses are due to conduction and ~ is a constant), but a stronger dependence on tem- perature (e.g., ~0 ~ z 4 representing radiation losses) diminishes the relative importance of upstream losses.

Equation (5.2) shows that for fast-flame branch, the residue of unburnt mixture behind the flame is transcendentally small in 0 (cf. (4.4)). This is actually a consequence of the first order kinetic scheme. If the reaction rate was proportional to (1 - I0n, n :~1 then the resi- due would be algebraically small in 0, for the same reason that Kassoy and Williams [5] get

algebraic perturbations in their large Dam- k6hler number study of fuel drop burning.

This work was supported partly by the U.S. Army Research Office and partly by the Na- tional Science Foundation under grant GP- 40965.

References 1. Adler, J., and Spalding, D. B., Proc. Roy. Soc. London

A261, 53 (1961).

2. Bush, W. B., and Fendell, F. E., Comb. Sci. and Tech. 1,

461 (1970).

3. Gerstein, M., and Stine, W. B., Fourteenth Symposium

(International) on Combustion, The Combustion Insti-

tute, Pittsburgh, PA, 1973, p. 1109.

4. Johnson, W. E., Arch. Rat ional Mechanics and

Analysis 13, 46 (1963).

5. Kassoy, D. R., and Willams, F. A., Phys. Fluids 11,

1343 (1968).

6. Simon, D. M., Belles, F. E., and Spakowski, A. E.,

Fourth Symposium (International) on Combustion,

. 1953, p. 126.

7. Spalding, D. B., Proc. Roy. Soc. London A240, 83 (1957).

8. Spalding, D. B., and Yumlu, V. S., Combust. Flame 3,

553 (1959).

9. Yumlu, V. S., Combust. Flame 10, 147 (1966).

10.Williams, F. A., Combust ion Theory, Addison-

Wesley, Palto Alto, CA, 1965.

Received 30 December 1974; revised2 September 1975